Various applications, such as non-contact optical metrology for semiconductor fabrication, require simulation of the optical response of two-dimensional gratings. The rigorous coupled wave analysis (RCWA) is one of the most important methods for such simulations Two-dimensional RCWA simulation has been considered by Soon Ting Han et al., Applied optics, vol. 31, no. 13, 2343-2352, 1992 and by Philippe Lalanne, J. Opt. Soc. Amer. Vol. 14, no. 7, 1592-1598, 1997. Although the RCWA is a powerful technique, the computational speed has been a serious issue, especially for 2D simulations. In an RCWA simulation, a finite number of diffraction orders are included in a calculation. For example, N orders may be included in a 1-D calculation, while NM orders may be included in a 2-D calculation (N orders in one direction and M orders in another direction). Thus 2-D RCWA calculations typically include a much larger number of orders than 1-D calculation, which undesirably increases computation time.
Accordingly, methods of reducing RCWA computation time for 2-D calculations are of interest. For example, U.S. patent application Ser. No. 11/305,449 by the present inventors considers symmetric 2-D cases where symmetry-induced relations among the diffraction orders are exploited to reduce computation time. However, further reduction in 2-D RCWA computation time is desirable.
In an RCWA calculation, the grating profile is expressed in terms of Fourier coefficients, one coefficient for each retained diffraction order. In the simple case of a grating having rectangular pillars, the Fourier coefficients have simple, well-known analytical expressions. For more complicated grating pillar shapes, standard methods for numerical evaluation of Fourier coefficients are applicable, although they can be time consuming. The time required for calculation of Fourier coefficients tends to be more significant for 2-D gratings, because typically many more Fourier coefficients are needed in a 2-D calculation than in a 1-D calculation. Discussions of the 2-D RCWA in the literature tend to treat the Fourier coefficients of the grating as given quantities (presumably from a numerical integration if necessary), and concentrate on refinements of other steps of the method. For example, Eero Nopomen et al. considers a 2D grating with circular pillars (J. Opt. Soc. Am. A/Vol. 11, No. 9, September 1994), but no details relating to how the Fourier coefficients are obtained is given.
Accordingly, it would be an advance in the art to provide an improved method for evaluating 2-D grating Fourier coefficients having reduced computational time requirements.